Bubbling phenomenon for semilinear Neumann elliptic equations of critical exponential growth

Lu Chen, Guozhen Lu*, Caifeng Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equations with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger–Moser exponential growth: {-dΔud+ud=ud(eud2-1)inΩ,∂ud∂ν=0on∂Ω, where d> 0 is a parameter, Ω is a smooth bounded domain in R2 , ν is the unit outer normal to ∂Ω . We first prove the existence of a ground state solution to the above equation. If d is sufficiently small, we prove that any ground state solution ud has at most one maximum point which is located on the boundary of Ω . The key point of the proof lies in proving that the distance between the maximum point Pd and the boundary ∂Ω satisfies d(Pd,∂Ω)≲d when d→ 0 and ud under suitable scaling transform converges strongly to the ground state solution of the limiting equation -Δw+w=w(ew2-1) . Our proof is based on the energy threshold of cut-off function, the concentration compactness principle for the Trudinger–Moser inequality, regularity theory for elliptic equation and an accurate analysis for the energy of the ground state solution ud as d→ 0 . Furthermore, by assuming that Ω is a unit disk, we remove the smallness assumption on d and show the maximum point of ground state solution ud must lie on the boundary of Ω for any d> 0 .

Original languageEnglish
Article number21
JournalCalculus of Variations and Partial Differential Equations
Volume63
Issue number1
DOIs
Publication statusPublished - Jan 2024

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